Analysis of Poly Phase Boost Converter Using PI Control Algorithms
Shiv Shanker Sharma^{1}, Alka Agrawal^{2}
^{1}M.Tech. II Year (Power System) Student
Institute of Technology & Managemnet, Bhilwara, Rajasthan, India
^{2}Asst. Professor, Department of Electrical Engineering
Institute of Technology & Managemnet, Bhilwara, Rajasthan, India
Abstract – This paper explains about the poly-phase boost converter and control algorithm which overcomes the problem of high ripple current in the tank capacitor. For the improvement of the functionality of the boost converter there are many methods available among which I consider PI controller in voltage mode control path. Initially I discussed the basic function of the boost converter. Then I derived the transfer function of the complete system. Then I considered model and simulate into matlab without PI controller. Finally I used PI controller in which the values of K_{p} and K_{i} has been derived using the Ziegler-nichols method and loop shaping method. Then I applied the control method on four phase boost converter. At last the output response of the both systems is compared and conclusion made upon that comparison.
Keywords -transfer function of boost converter; closed loop system transfer function; Zieglar-Nichols methodt; Loop shaping method Parallel operation, Poly-phase converter
I. INTRODUCTION
In designing DC converters, parameters such as ratio of energy stored in inductor and capacitor to energy delivered to load in one period, maximum current in the switch and the value of the RMS current in the output capacitor have great importance and it is necessary to be considered. The motivation for this work is expressed through consideration of the above parameters in per unit measured for the two basic converters namely the buck and the boost converter [1]. Consider the boost converter in Fig.1 with per unit values defined as.
V_{dc}=1, D=0.5, T_{s}=1, E_{0}=1, P_{o}=1, ∆I_{o}/I_{L}=20%, ∆V_{o}/V_{o}=1%
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Fig 1 Buck & Boost Converter respectively
Table I gives the reactive elements and their energy storage capacity for the basic converters. From the table it is obvious that the boost converter requires total energy storage far in excess of buck converter
Table. I: Comparing Buck & Boost Converters (per unit values)
L (p.u.) C (p.u.) V (p.u.) (p.u.) (p.u.) BOOST 2.5 12.5 2 1.25 25 BUCK 1.25 10 0.5 1.25 1.25
One-way of reducing the storage requirement is increasing the switching frequency however this is not practicable in all instances. During the on state of the switch, the capacitor has to supply the entire load current in the boost converter; this discontinuity of current in the capacitor increases the RMS value of current and also increases the amount of capacitor which is needed to keep the ripple voltage low.
The power dissipation in the ESR of the capacitor is also high. In standard designs it is not uncommon to see tank capacitors one or two orders of magnitude higher than the ideally required capacitance A way to overcome this problem is using poly-phase operation with appropriate phase shift in the control circuit of main switches [2,3]. Fig 2 shows such a poly-phase boost converter (N=4). Fig. 3 shows the conduction intervals of the four switches in the converter. It is seen that at any time at least one of the converters is supplying the load in addition to the capacitor.
The frequency of ripple current in the output capacitor is N times compared to the single stage and therefore the value of the capacitor required can be reduced. The same circuit topology is also applicable to UPF rectifiers [4-6]
In such a scheme, the following advantages are obvious.
• Output capacitor is rated for lower ripple current and higher ripple frequency (nfs).
• Source current has higher ripple and at higher frequency (nfs).
• Another no obvious advantage is that the multi-phase converter may be operated with less number of stages when the load current is low. This will lead to operation under CCM at light load as well as better efficiency
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Fig 2 Multi-phase boost Converter(N=4)
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Fig 3 The case with Dmax ≥ 1-1/4
II. MATHEMATICAL MODELING
Transfer Function of Boost Converter - Basic circuit of the boost converter is shown in Figure 4
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Fig 4 Basic boost converter
Here, L is the inductor, C is the output capacitor and R_{L} is the resistor which is consider as a load. I_{L} is the current flow through the circuit. Switch is triggered by the pulse which is generated by PWM technique. Switch remains on during T_{on} cycle and off during T_{off} cycle so triggering is depends on the duty cycle. V_{dc} is the D.C. input voltage supply which is taken from the bridge rectifier which converts A.C. input voltage into D.C., V_{o} is the output of the boost converter which is larger than the input voltage V_{dc}.
Now to achieve proper objective of converter, it is need to measure and maintain output voltage at required voltage level. So for that purpose it is needed to use feedback loop into the system that is shown in Fig 5
By technique of averaging and linearising small signal model around a operating point the Control-to-output, input to output voltage transfer function for open loop boost converter are obtained as:
[IMG]file:///C:\Users\John\AppData\Local\Temp\msohtmlclip1\01\clip_image014.gif[/IMG](1)
[IMG]file:///C:\Users\John\AppData\Local\Temp\msohtmlclip1\01\clip_image016.gif[/IMG]…(2)
[IMG]file:///C:\Users\John\AppData\Local\Temp\msohtmlclip1\01\clip_image018.gif[/IMG]
Fig 5 Single Phase Closed Loop System of Converter
Fig 5 shows the close loop control scheme of a boost converter using PI algorithm. Here V_{o} is compared to V_{ref}. The error signal of the comparator is processed by PI controller and voltage control signal is sent to PWM block, which eventually produce duty ratio. Then it is added with V_{in} which is given to the system
III. SIMULATION
For the simulation purpose I considered the following model:
Input Voltage (dc) :24 volt
Output Voltage (dc) :48 Volt
Boost Inductor (L) :100 mH
Rated Power :16 W
Switching Frequency :1 kHz
Normally, duty cycle for boost converter is considered in between 0.5 to 1. Selection of duty cycle depends on input voltage supply and required output voltage [1]. When boost converter is used without using PI controller it gives steady state error of 25%. So I used PI controller to improve the performance of boost converter. To find out the value of K_{p} and K_{i} , I used Ziegler-Nichols step response method and Loop shaping method [4, 5].
Applying step function to the system and analyzing its output response, I got two parameters L = 1 and T = 0.004
Using these, the value of K_{p} and K_{i} can be found by Ziegler-Nichols method which is given below.
K_{p} = 0.0036 and K_{i}= 3.33
Now, applying these values into PI controller of the closed loop system and simulate it into the matlab I got the response as shown in fig. 6.
From fig. 6 it is shown that it removes steady state error but initially it provide high oscillations.
Now, using the Loop shaping method, eq. 9 and parameter of the considered model I got the following two relation [4 ,5].
2 ∗ξ * ω =(3000 * K_{p} − 750………………(3)
3000 * K_{i} = ω^{2}……………………………...(4)
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Fig. 6 Output Response of system using Ziegler-Nichols Method
For PI controller, ξ is maintained at 0.7 and parameters Kp and Ki must be a larger [6]. So, using eq.3 and eq.4
Kp = 2.68 and Ki= 13146
Now, applying these values into PI controller of the closed loop system and simulate it into the matlab I got the response as shown in fig.7.
From the fig. 7 it is shown that it reduce steady state error and it doesn’t produce any oscillation which was presented during the Ziegler-Nichols method. The performance of the system is improved by the PI controller using loop shaping method
[IMG]file:///C:\Users\John\AppData\Local\Temp\msohtmlclip1\01\clip_image022.gif[/IMG]
Fig. 7 Output voltage of close loop single phase boost converter
From fig. 7 it is revealed that transient time is 4ms, steady state error is 1.04% and ripple are 5%.
Now the same PI algorithm I applied to proposed four phase boost covereter. Here four PWM gate signal shifted by 90 degrees phase from each other and a single common duty ratio is imposed on the switches, I got the response as shown in fig 8
[IMG]file:///C:\Users\John\AppData\Local\Temp\msohtmlclip1\01\clip_image024.jpg[/IMG]
Fig 8 Output voltage of close loop four phase boost converter
Fig. 8 revealed that steady state error is juat 0.0625%., transient response time is only of 1µs. No overshoot and output voltage ripple is less than 0.1%
IV. COMPARISON OF SINGLE PHASE AND FOUR PHASE BOOST CONVERTERS
S.No Features Single Phase Boost Conv. Poly Phase Boost Conv. 1 Inductor size A single inductor of same
RatingSame inductance required divided in N parts for N phase 2 Storage requirement High Reduces by factor 1/N^{2} 3 Frequency of operation Low (f) High (Nf) 4 Power dissipation in ESR High Low 4 Efficiency Low High 5 Output ripple High Low 6 Mode of operation Only DCM with low load current CCM is possible even with low load current, thus less stages of operation 7 Steady state error High Low 8 Transient time Of the order of mili seconds Of the order of nano seconds 9 Overshoot High (up to 40%) Low (maximum 0.1%)
V. CONCLUSION
This paper presents analysis of the boost converter using Ziegler-Nichols method and loop shaping method. From the above result, following are the conclusion that can be drawn from this paper.
V. FUTURE WORK
- Using the boost converter without PI controller, it produces steady state error of 25%.
- Boost converter used with PI controller applying Ziegler-Nichols method removes steady state error after 2.5 sec. But it produce high oscillation and maximum peak overshoot of 900% that shown in fig. 6. It also produced 14.5% of output ripple. So it is undesirable.
- Boost converter used with PI controller applying loop shaping method removes steady state error faster and also removes oscillation which is shown in fig. 7. It also produced only 0.5% output ripple which is lower than the Ziegler-Nichols method.So, from above conclusion can be made that loop shaping method gives better response than the Ziegler-Nichols method for the proposed model
- On comparing the performance of single phase and four phase boost converters we found that steady state as well as transient performance of multi phase converter is much superior to single phase. Besides having same size inductor and small size storage multi phase converter can be operated with reduced input current ripple and less output voltage ripple
A fully digitized implementation of the proposed control methods should be carried out through the development of FPGA. Further the development of application specific integrated circuit (ASIC) should be explored to provide a smaller, more reliable and cheaper controllers. Further, scope of future work is to implement the open loop and closed loop control of Multi-phase Boost Converter using the DSPTMS320LF2407
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[5] Copeland, Brain R. "The Design of PID Controllers using Ziegler Nichols Tuning," 2008.
[6] D.P.Eckman, Automatic Process Control. New Delhi: Wiley Eastern,1992
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